where \(\mathcal {S}_{n} = \lbrace  s: n^{-1/2}s + \hat{\theta }_n \in T(\epsilon )\rbrace \) , and \(T(\epsilon )\)  is the \(\epsilon \) -tubular neighborhood around \(\mathcal {M}\)  (see, for instance, [1]} for an explanation of the tubular neighborhood), chosen to be closed. For the purposes of this proof, we require that \(\mathcal {I}^1_n(\beta )\)  go to zero in the dual limit, and that the limits with respect to \(n\)  and with respect to \(\beta \)  be interchangeable. This fact is established in supporting Lemma REF .
